The blocks and weights of finite special linear and unitary groups

Abstract

This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $S{L}_n(\epsilon q)$ in the non-defining characteristic $\ell \ge 3$. Secondly, we describe how the $\ell$-weights of $S{L}_n(\epsilon q)$ can be obtained from the $\ell$-weights of $G{L}_n(\epsilon q)$ when $\ell \nmid \gcd (n,q-\epsilon )$, and verify the Alperin weight conjecture for $S{L}_n(\epsilon q)$ under the condition $\ell \nmid \gcd (n,q-\epsilon )$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $\ell$-block of $S{L}_n(\epsilon q)$ if $\ell \nmid \gcd (n,q-\epsilon )$.

revised version, to appear in Journal of Algebra